Verification of high-order mixed finite-element solution of transient magnetic diffusion problems

We develop and present high-order mixed finite-element discretizations of the time-dependent electromagnetic diffusion equations for solving eddy-current problems on three-dimensional unstructured grids. The discretizations are based on high-order H(Grad), H(Curl), and H(Div) conforming finite-element spaces combined with an implicit and unconditionally stable generalized Crank-Nicholson time differencing method. We develop three separate electromagnetic diffusion formulations, namely the E (electric field), H(magnetic field), and the A-φ (potential) formulations. For each formulation, we also provide a consistent procedure for computing the secondary variables J(current flux density) and B(magnetic flux density), as these fields are required for the computation of electromagnetic force and heating terms. We verify the error convergence properties of each formulation via a series of numerical experiments on canonical problems with known analytic solutions. The key result is that the different formulations are equally accurate, even for the secondary variables J and B, and hence the choice of which formulation to use depends mostly on relevance of the natural and essential boundary conditions to the problem of interest. In addition, we highlight issues with numerical verification of finite-element methods that can lead to false conclusions on the accuracy of the methods.